Equity Premium: - Does it exist? Evidence from Germany and...

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Equity Premium: - Does it exist? Evidence from Germany and United Kingdom

Michael E. Drew Mirela Mallin

Tony Naughton and

Madhu Veeraraghavan

Discussion Paper No. 170, January 2004

Series edited by Associate Professor Andrew C Worthington

School of Economics and Finance

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Equity Premium: Does it exist? Evidence from Germany

and United Kingdom

Michael E. Drew Mirela Mallin

Tony Naughton and

Madhu Veeraraghavan

School of Economics and Finance Queensland University of Technology GPO Box 2434 Brisbane Queensland 4001 Australia School of Accounting and Finance Griffith University PMB 50 Gold Coast Mail Centre Gold Coast Queensland 9726 Australia School of Economics and Finance RMIT City Campus GPO Box 2476V Melbourne 3001 Australia Department of Accounting and Finance The University of Auckland Business School Private Bag 92019, Auckland, New Zealand

Abstract

Malkiel and Xu (1997) state that idiosyncratic volatility is highly correlated with size

and that it plays a powerful role in explaining expected returns. In this paper we ask

(a) whether idiosyncratic volatility is useful in explaining the variation in expected

returns; and, (b) whether our findings can be explained by the turn of the year effect.

We find that (a) our three-factor model provides a better description of expected

returns than the CAPM. That is, we find that firm size and idiosyncratic volatility are

related to security returns. In addition, we also find that our findings are robust

throughout the sample period. We show that the CAPM beta alone is not sufficient to

explain the variation in stock returns.

JEL Classification: G110, G120, G150

Keywords: Idiosyncratic Volatility, Size Effect, CAPM, Risk Premia

1. Introduction

Why has the rate of return on equities been higher than the rate of return on risk free

assets? The question first posed by Prescott and Mehra (1985) has been termed the

“equity premium puzzle”. One simple answer to this challenging question is that

equities are more risky than bonds and thus investors require a premium for taking

this additional risk. In the context of the Capital Asset Pricing Model [henceforth

CAPM] high beta stocks generate superior returns since there’s a linear relationship

between the stock’s beta and the expected return. However, recent tests show that

the cross-section of average stock returns shows little or no relation to the market

betas of the CAPM.

The results indicate that variables such as firm size1, leverage, firm’s book value of

equity to its market value, and more recently idiosyncratic volatility adequately

explain the cross-section of average stock returns better than the beta of a stock. In

an important paper Malkiel and Xu (1997) confirm the controversial finding of Fama

and French [hereafter FF] (1992) that beta does not appear as an explanatory

variable when attempting to model the annual returns on US stocks from 1963

through 1990.

They find that portfolios of smaller firms produce risk-adjusted rates of return that are

greater than the returns from portfolios of larger firms. Interestingly, they report that

1 Banz (1981) and Reinganum (1981) show that risk-adjusted stock returns are a monotonically

decreasing function of firm size. Banz (1981) shows that going long in a portfolio of small firms and

going short in a portfolio of big firms generates excess returns of approximately 20 percent per year.

Schultz (1983) shows that investors can earn risk-adjusted returns after transaction costs by holding

small firms for short periods. Also, see, Schwert (1983), Lakonishok and Smidt (1983), Chelley-Steeley

and Steeley (1996), Fletcher (1997), Priestley (1997), Heston et al (1999), Charitou et al (2001), Dimson

and Marsh (2001), Beltratti and Massimo (2002) and Dissanayake (2002).

idiosyncratic volatility is highly correlated with firm size and that it plays an important

role in explaining expected returns. That is, they observe that portfolios of smaller

companies have higher idiosyncratic volatility and thus these portfolios post

significantly higher average returns suggesting that asset returns are influenced by

factors that are not related to economic conditions. Finance theory states that

through the process of diversification “idiosyncratic factors” can be cancelled out and

thus asset returns are only influenced by systematic factors. In this article, we

advance this argument by providing out of sample evidence from two European stock

markets – Germany and United Kingdom.

We specifically ask:

(a) Is idiosyncratic volatility needed to explain the variation in average stock returns?

and,

(b) How are firm size and idiosyncratic volatility related to security returns?

We ask these two questions since recent research suggests that firm size is strongly

related to idiosyncratic volatility (Malkiel and Xu, 1997). Malkiel and Xu (1997) report

that portfolios of smaller stocks tend to have larger idiosyncratic volatility than

portfolios of larger stocks. More importantly, they show that idiosyncratic volatility is

highly correlated with firm size and that it plays a powerful role in explaining the

cross-section of expected returns. Malkiel and Xu (2000) report that idiosyncratic

volatility affects returns even after controlling for firm size and book-to-market equity

effects. They state that idiosyncratic volatility will affect asset returns when not every

investor is able to hold the market portfolio. Campbell, Lettau, Malkiel and Xu (2001)

find a noticeable increase in firm level volatility relative to the market volatility. Their

results indicate that firm specific volatility is the largest component of the total

volatility of an average firm. Xu and Malkiel (2001) report that volatility is associated

with the level of institutional ownership as well as a positive relationship between

idiosyncratic volatility and expected earnings growth. Drew and Veeraraghavan

(2002) show that small and high idiosyncratic volatility stocks generate superior

returns in Hong Kong, India, Malaysia and Philippines. Their findings support Malkiel

and Xu (1997 and 2000) who document that idiosyncratic risk is useful in explaining

the cross-section of expected returns.

Interestingly, Drew, Naughton and Veeraraghavan (2003) find that small and low

idiosyncratic volatility firms generate superior returns than big and high idiosyncratic

volatility firms for equities listed in Shanghai Stock Exchange. They propose a

behavioral explanation in that they forward irrational investor behavior as a possible

explanation in the spirit of Thaler (1999), Daniel and Titman (1999) and Hirshleifer

(2001). They conclude that Chinese investors are quasi-rational investors in the

sense of Thaler (1999).

Hamao, Mei and Xu (2002) state that the role of idiosyncratic risk in asset pricing has

largely been ignored since standard finance theory argues that only systematic risk

should be priced in the market. In a similar vein, Xu and Malkiel (2003) observe that

the behavior of idiosyncratic volatility has received far less attention in the finance

literature. This is because standard finance theory argues that idiosyncratic volatility

can be eliminated in a well-diversified portfolio. Barber and Odean (2000) and

Benartzi and Thaler (2001) report that both individual investors’ portfolios and mutual

fund portfolios’ are undiversified. Goyal and Santa-Clara (2001) argue that the lack of

diversification suggests that the relevant measure of risk for many investors may be

the total risk. It is important to note that little, if any, has been published on whether

idiosyncratic volatility can explain the cross section of expected stock returns.

In light of these discussions we investigate whether idiosyncratic volatility can serve

as a useful proxy for systematic risk and whether it helps explain the variation in

average stock returns for equities listed in German and United Kingdom markets.

The rest of the paper is organized as follows. Section 2 describes the data and

methodology employed in this paper. Section 3 presents our findings while Section 4

presents concluding comments.

2. Data and Methodology

2.1 Data and the model

We obtain monthly stock returns and market values of all listed firms in Germany and

United Kingdom covering the period 1991 to 2001 from DataStream. The relationship

between stock returns, overall market factor, size (ME), and idiosyncratic volatility is

investigated by employing the following model.

Rpt – Rft = apt + bp (Rmt-Rft) + spSMBt + hpHIVMLIVt +εpt [1]

Where, Rpt is the average return of a portfolio (S/L, S/M, S/H; B/L, B/M and B/H)2. Rft

is the risk-free rate3 observed at the beginning of each month. Market, is long the

market portfolio and short the risk free asset; SMB, is long small capitalization stocks

and short large capitalization stocks; HIVMLIV, is long high idiosyncratic volatility

2 S/L Portfolio = Small firms with low idiosyncratic volatility

S/M Portfolio = Small firms with medium idiosyncratic volatility

S/H Portfolio = Small firms with high idiosyncratic volatility

B/L Portfolio = Big firms with low idiosyncratic volatility

B/M Portfolio = Big firms with medium idiosyncratic volatility

B/H Portfolio = Big firms with high idiosyncratic volatility

3 We use the Germany Benchmark bond 10-year yield for Germany and the 1-month interbank rate for

United Kingdom as risk-free rate of return.

stocks and short low idiosyncratic volatility stocks. The factor loadings bp, sp and hp

are the slopes in the time-series regression.

2.2. Methodology

In this paper we follow the mimicking portfolio approach of FF (1996) in constructing

portfolios on firm size and idiosyncratic volatility. We follow this approach since

Malkiel and Xu (1997 and 2000), Xu and Malkiel (2001), Drew and Veeraraghavan

(2002b) and Drew, Naughton and Veeraraghavan (2003) suggest that idiosyncratic

volatility may be relevant for asset pricing and that it may serve as a useful proxy for

systematic risk.

Size Portfolios

At the end of December of each year t stocks are assigned to two portfolios of size

(Small or Big) based on whether their December market equity (ME) [defined as the

product of the closing price times number of shares outstanding] is above or below

the median ME. We form portfolios as of December of each year since most firms in

Germany have December as fiscal year end. For firms listed in United Kingdom size

portfolios are constructed at the end of March of each year since most firms have

March as fiscal year end.

Idiosyncratic Volatility Portfolios

In an independent sort the same stocks are allocated to three idiosyncratic volatility

portfolios (Low, Medium, and High) based on the breakpoints for the bottom 33.33

percent and top 66.67 percent. We first compute the variance of returns for each

stock in the sample. We define the variance of returns as the total risk of a stock. We

then estimate the beta for each stock by using the covariance / variance approach.

We define systematic risk as the beta of a stock multiplied by the variance of the

index. Note that we require the previous 24 months of average returns to calculate

the variance or beta of the stock. Stocks that do not have 24 months of continuous

returns are excluded from the sample. Similarly, we use the previous 24 months of

market returns to calculate the variance of the index. We define idiosyncratic volatility

as the difference between total risk and the systematic risk of a stock.

Six Intersection and three zero investment portfolios

We form six intersection and three zero investment portfolios. The six intersection

portfolios formed are (S/L, S/M, and S/H; B/L, B/M, and B/H). The three zero

investment portfolios are RMRFT, SMB and HIVMLIV. We define the three zero

investment portfolios RMRFT, SMB, and HIVMLIV as follows: RMRFT is long the

overall market portfolio and short the risk free asset. SMB (Small minus Big) is the

difference each month between the average of the returns of the three small stock

portfolios (S/L, S/M, and S/H) and the average of the returns of the three big

portfolios (B/L, B/M, and B/H). HIVMLIV (High Idiosyncratic Volatility minus Low

Idiosyncratic Volatility) is the difference between the average of the returns of the two

high idiosyncratic volatility portfolios (S/H, B/H) and the average of the returns on the

two low idiosyncratic volatility portfolios (S/L, B/L).

3. Results

3.1. Performance of the Intersection and Zero Cost Portfolios

Germany

Table 1.0 Sample Characteristics - Germany

Number of Companies in Portfolios Formed on Size and Idiosyncratic Volatility 1993 to 2001

YEAR S/L S/M S/H B/L B/M B/H Total

1993 16 21 9 37 12 7 102

1994 14 24 10 42 12 2 104

1995 16 23 9 45 8 4 105

1996 16 20 15 44 10 3 108

1997 14 22 16 43 13 5 113

1998 22 19 12 44 16 4 117

1999 22 26 5 42 23 3 121

2000 22 31 5 40 30 4 132

2001 29 26 15 40 38 7 155

Average 19 24 11 42 18 4 117

Table 1, reports the average numbers of firms in each portfolio for the sample period.

B/L portfolio has an average of 42 firms followed by the S/M portfolio with an average

of 24 firms. The S/L and B/M portfolios have an average of 19 and 18 firms

respectively. The least number of firms are in S/H and B/H portfolios with an average

of 11 and 4 respectively. Our first research question is to investigate whether a

multifactor asset-pricing model explains the cross-section of average stock returns.

Specifically, this study is interested in whether an overall market factor, firm size and

idiosyncratic volatility can explain the cross-sectional pattern of stock returns. The

mean monthly returns and the regression parameters are reported in Table 2.

Table 2.0

Summary Statistics and Multifactor Regressions for Portfolios Formed on Size and Idiosyncratic Volatility - Germany

1993-2001 Summary Statistics


Size Low Medium High Low Medium High

Panel A: Summary Statistics

Means Standard Deviations

Small 0.46 0.83 1.61 3.94 4.45 4.92

Big 0.52 0.76 1.10 4.38 3.73 8.38

Table 2, Panel A, shows the summary statistics while Panel B shows the regression

coefficients of the three-factor model. Our results show that all six portfolios generate

positive returns with the S/H portfolio generating the highest return of 1.61 per cent

per month. The overall performance of the six portfolios is graphically shown in figure

1.0. Our findings also show that the overall market factor generates a return of 0.52

per cent per month while the other two mimic portfolios, SMB and HIVMLIV generate

a return of 0.17 per cent per month and 0.87 per cent per month respectively. Since,

the mimic portfolios for size and idiosyncratic volatility generate superior returns; we

argue that this is a compensation for risk not captured by the CAPM. That is, we

advance a risk-based explanation and suggest that small and high idiosyncratic

volatility firms are riskier than big and low idiosyncratic volatility firms.

Table 2 - Continued Multifactor Regressions for Portfolios Formed on Size and Idiosyncratic Volatility

Regression Coefficients



Panel B: Rpt – Rft = apt + bp (Rmt-Rft) + spSMBt + hpHIVMLIVt +εit

a t (a)

Small 0.000 0.003 0.002 0.353 1.106 1.352

Big 0.002 0.004 0.000 1.028 1.898 0.076

b t(b)

Small 0.541 0.587 0.680 11.626 11.205 17.803

Big 0.708 0.531 0.569 18.377 12.250 10.172

s t(s)

Small 0.311 0.454 1.349 4.751 6.161 25.111

Big 0.103 -0.052 -0.935 1.899 -0.856 -11.874

h t(h)

Small 0.037 0.145 0.853 0.712 2.458 19.807

Big -0.097 0.047 1.086 -2.239 0.957 17.213

R2 s(e)

Small 0.65 0.68 0.91 2.57 2.89 2.11

Big 0.76 0.69 0.86 2.13 2.39 3.09

DW

Small 1.96 1.96 1.99

Big 1.92 1.93 1.98

Panel B, shows, that the intercept, is statistically insignificant and close to zero for all

six portfolios. The findings also show that the b coefficient is positive and highly

significant for the six portfolios. The s coefficient increases monotonically and is

positive and highly significant for the three small stock portfolios. As far as three big

portfolios are concerned the s coefficient is positive for B/L but negative for B/M and

B/H portfolios.

Note that our findings are consistent with that of FF (1996) who argues that small

firms load positively on SMB while big firms load negatively on SMB. The h

coefficient increases monotonically for all six portfolios and is highly significant at the

1% level for S/H and B/H portfolios. The other portfolios display low levels of

statistical significance. We do not find any evidence of autocorrelation since the d-

statistic close to 2 for all six portfolios. Similarly, the test for multicollinearity shows

no evidence of multicollinearity between the independent variables.

Insert Figure 1.0 about here

United Kingdom

Table 3.0

Sample Characteristics – United Kingdom Number of Companies In Portfolios Formed on Size and Idiosyncratic Volatility

1993 to 2001 YEAR S/L S/M S/H B/L B/M B/H Total

1993 41 130 204 204 117 40 736

1994 36 128 207 214 125 41 751

1995 39 113 215 218 149 41 775

1996 40 134 209 241 152 71 847

1997 68 148 215 239 164 93 927

1998 101 144 242 246 208 102 1043

1999 138 178 241 248 213 137 1155

2000 140 198 252 273 224 149 1236

2001 134 187 285 295 257 133 1291

Average 82 151 230 242 179 90 973

Table 3, reports the average number of firms in each portfolio for the sample period.

The B/L portfolio has the largest number of firms with an average of 242, followed

closely by the S/H portfolio with an average of 230 firms. The S/M portfolio contains

an average of 151 firms while B/M contains an average of 179 firms. The S/L and

B/H portfolios have an average of 82 and 90 firms respectively. In Table 4.0 we

report the summary statistics and regression coefficients of our multifactor model.

Panel A, shows, the summary statistics while Panel B shows the regression

coefficients.

Table 4.0

Summary Statistics and Multifactor Regressions for Portfolios Formed on Size and Idiosyncratic Volatility – United Kingdom

1993-2001 Summary Statistics



Panel A: Summary Statistics

Means Standard Deviations

Small -0.18 -0.01 1.16 2.02 3.07 6.91

Big 0.79 0.18 3.36 4.09 3.40 8.89

Our results show that with the exception of two portfolios all other portfolios generate

positive returns. Our results also show that the B/H portfolio generates the highest

return of 3.36 per cent per month while the S/H portfolio generates a return of 1.16

per cent per month. Our findings for United Kingdom differ in this respect with that of

Germany where we found that the small and high idiosyncratic volatility portfolios

generate the highest returns.

The overall performance of the six portfolios is graphically shown in figure 2.0. Our

findings also show that the overall market factor generates a mean monthly return of

0.32 per cent per month while the mimic portfolio for size and idiosyncratic volatility

generate a return of –1.46 per cent per month and 1.96 per cent per month

respectively. Thus, in the case of United Kingdom we document a big firm effect.

Note that in Germany we found a small firm effect. However, it is to be noted that in

both the markets investigated in this paper we document an idiosyncratic volatility

effect. That is, portfolios with high idiosyncratic volatility firms generate higher returns

than portfolios with low idiosyncratic volatility firms.

Table 4 - Continued

Multifactor Regressions for Portfolios Formed on Size and Idiosyncratic Volatility Regression Coefficients



Panel B: Rpt – Rft = apt + bp (Rmt-Rft) + spSMBt + hpHIVMLIVt +εit

a t (a)

Small -0.002 -0.004 0.001 -1.444 -1.464 0.385

Big -0.000 -0.001 0.004 -0.171 -0.299 2.200

b t(b)

Small 0.306 0.391 0.549 5.949 5.331 7.196

Big 0.525 0.440 0.281 8.459 5.976 5.378

s t(s)

Small 0.106 0.101 0.714 1.129 0.754 5.129

Big -0.452 -0.565 -1.063 -3.989 -3.495 -11.148

h t(h)

Small 0.089 0.225 0.975 3.479 6.161 25.612

Big 0.004 0.167 1.118 0.123 3.793 42.925

R2 s(e)

Small 0.72 0.65 0.88 2.63 2.32 2.41

Big 0.67 0.69 0.96 1.96 2.39 1.65

DW

Small 1.99 1.98 1.96

Big 1.97 2.07 1.96

In Table 4, Panel B, we report the coefficients of our multifactor model. Our findings

show that the intercept, a coefficient, is indistinguishable from zero for all six

portfolios. The b coefficient is positive and statistically significant for all portfolios.

The s coefficient is positive for the three small stock portfolios and statistically

significant only for S/H portfolio, while the big stock portfolios show negative

coefficients with statistical significance. The h coefficient increases monotonically for

all six portfolios and is highly significant at the 1% level for five out of six portfolios.

As far as the diagnostics are concerned we find no evidence of autocorrelation or

multi-collinearity in our sample.


3.2 Results from turn of the year effect

Germany

Prior research on the behaviour of stock prices documents a strong seasonality effect

occurring in the month of January, especially for small size stocks. This effect has

been described as the January effect. Research also shows that monthly seasonality

is linked to the size of the firm. Therefore, a natural extension to the size effect is to

examine whether it displays monthly seasonality. Thus, we ask whether multifactor

models findings can be explained by the turn of the year effect. In this model we add

a dummy variable that takes the value “1” for the month of January and “0” for

remaining months. Our model takes the following form:

Rpt – Rft = αpt + bp (Rmt-Rft) + spSMBt + hpHIVMLIVt + γpDJANt+ εt

Table 5.0

Tests for Turn of the Year Effect – Germany



Rpt – Rft = apt + bp (Rmt-Rft) + spSMBt + hpHIVMLIVt + γp Jant+ εit

a t (a)

Small 0.001 0.003 0.002 0.377 1.214 1.293

Big 0.002 0.004 0.000 1.018 1.940 0.130

b t(b)

Small 0.542 0.589 0.680 11.538 11.172 17.650

Big 0.708 0.532 0.570 18.230 12.195 10.103

s t(s)

Small 0.309 0.446 1.349 4.613 5.919 24.518

Big 0.102 -0.057 -0.938 1.830 -0.923 -11.636

h t(h)

Small 0.037 0.146 0.853 0.711 2.461 19.707

Big -0.097 0.047 1.087 -2.225 0.963 17.133

γ t(γ)

Small -0.001 -0.005 -0.000 -0.138 -0.551 0.853

Big -0.000 -0.003 -0.002 -0.129 -0.448 -0.200

R2 s(e)

Small 0.57 0.59 0.91 2.58 2.90 2.12

Big 0.76 0.59 0.86 2.14 2.40 3.10

DW

Small 1.97 1.95 1.98

Big 2.05 1.94 2.02

Table 5, shows the regression coefficients for the multifactor model. Our findings do

not reveal any evidence of the turn of the year effect for Germany since the

coefficient for the January dummy is not statistically significant for any of the six

portfolios. Thus, we reject the claim that the multifactor model results can be

explained by the seasonality effect.

United Kingdom

In the case of United Kingdom we test for both January and April effects. In this

model January dummy is represented by γ while April dummy is represented by θ.

Our time-series model takes the following form:

Rpt – Rft = αpt + bp (Rmt-Rft) + spSMBt + hpHIVMLIVt + γpDJANt+ θpDAPRILt + εpt

Table 6

Tests for Turn of the Year Effect (January and April) – United Kingdom



Rpt – Rft = apt + bp (Rmt-Rft) + spSMBt + hpHIVMLIVt + γp Jant+ θpFebt+εit

a t (a)

Small -0.002 -0.003 0.000 -0.950 -1.243 0.113

Big -0.000 -0.001 -0.003 -0.331 -0.365 -1.524

b t(b)

Small 0.312 0.396 0.557 5.969 5.274 7.194

Big 0.530 0.450 0.284 8.389 4.986 5.406

s t(s)

Small 0.100 0.100 0.731 1.063 0.738 5.216

Big -0.442 -0.552 -1.075 -3.862 -3.383 11.310

h t(h)

Small 0.087 0.225 0.981 3.361 6.047 25.538

Big 0.007 0.172 1.114 0.237 3.837 42.744

γ t(γ)

Small -0.004 -0.001 0.011 -0.801 -0.119 1.283

Big 0.006 0.008 -0.009 0.946 0.778 -1.551

θ t(θ)

Small -0.006 -0.0047 -0.000 -1.080 -0.461 -0.072

Big 0.006 0.008 -0.009 0.946 0.778 -1.551

R2 s(e)

Small 0.67 0.69 0.88 1.63 2.34 2.42

Big 0.64 0.66 0.92 1.97 2.82 1.64

DW

Small 1.99 1.98 1.97

Big 1.98 2.07 1.96

Once again, our findings reveal no evidence of the turn of the year effect since the

January and April dummy are statistically significant for any of the six portfolios.

Thus, we argue that the multifactor model is robust throughout the sample period.

We also do not find any evidence of autocorrelation or multicollinearity in our sample.

3.3 Factors of risk and risk premia

Germany

Table 7.0 Market, Size and Idiosyncratic Volatility Premia – Germany

Portfolio Market Premium

(%)

Size

Premium (%)

Idiosyncratic Volatility

Premium (%)

S/L 0.28 (11.626)

0.05 (4.751)

0.032 (0.712)

S/M 0.30 (11.205)

0.07 (6.161)

0.39 (2.458)

S/H 0.35 (17.803)

0.22 (25.111)

0.74 (19.807)

B/L 0.36 (18.377)

0.01 (1.899)

-0.08 (-2.239)

B/M 0.27 (12.250)

-0.00 (-0.856)

0.04 (0.957)

B/H 0.29 (10.172)

-0.15 (-11.874)

0.94 (17.213)

Our findings show that the market portfolio generates positive risk premia for all six

portfolios. We find that the (B/L) portfolio generates the highest risk premia of 0.36

percent per month (t-statistic = 18.377). We also report that idiosyncratic volatility is

highly correlated with firm size. Once again, we find that the (S/H) portfolio generates

the highest size premium of 0.22 per cent per month (t-statistic = 25.111) while the

(B/H) portfolio generates the highest idiosyncratic volatility premia of 0.94 per cent

per month (t-statistic = 17.213). We also observe that the premia associated with

idiosyncratic volatility increases monotonically for the three small and big stock

portfolios. As, small and high idiosyncratic volatility firms generate higher risk premia

we argue that these factors are compensation for the risk missed by the CAPM. Once

again our findings are consistent with that of Malkiel and Xu (1997 and 2000). Our

results are summarized in Figure 3.0.


United Kingdom

Table 8.0

Market, Size and Idiosyncratic Volatility Premia – United Kingdom

Portfolio Market

Premium (%)

Size

Premium (%)

Idiosyncratic Volatility

Premium (%)

S/L 0.09 (5.949)

-0.15 (1.129)

0.17 (3.479)

S/M 0.12 (5.331)

-0.15 (0.754)

0.44 (6.161)

S/H 0.17 (7.196)

-1.04 (5.129)

1.91 (25.612)

B/L 0.16 (8.459)

0.65 (-3.989)

0.01 (0.123)

B/M 0.14 (4.976)

0.82 (-3.495)

0.33 (3.793)

B/H 0.08 (5.378)

1.55 (-11.148)

2.19 (42.925)

Our findings reveal that the market factor generates positive risk premia for all six

portfolios. As with Germany we find that the (S/H) portfolio generates the highest risk

premia of 0.17 percent per month (t-statistic = 7.196). Interestingly, our findings for

United Kingdom are different from that of Germany in that we document a big firm

effect in UK. This is because we find that the three small stock portfolios generate

negative risk premia while the three big stock portfolios generate positive risk premia.

We observe that the (B/H) portfolio generates the highest size premia of 1.55 percent

per month (t-statistic = 11.148). As far as idiosyncratic volatility premia is concerned

we see a monotonic increase for all six portfolios. We find that the (B/H) portfolio

generates the highest premia of 2.19 percent per month (t-statistic = 42.925) followed

by the (S/H) portfolio of 1.91 percent per month (t-statistic = 25.612). The findings in

this respect are consistent with that of Germany. We suggest that if investors are

willing to take additional risks they should invest in firms with such characteristics.

We summarize these results in Figure 4.0.


4. Conclusions

The Capital Asset Pricing Model states that expected returns on securities are a

positive linear function of their market betas. However, Malkiel and Xu (1997 and

2000) contradict the CAPM by observing that idiosyncratic volatility is priced in the

market and hence related to stock returns. In this paper we ask (a) whether

idiosyncratic volatility is correlated with firm size and is it useful in explaining the

variation in stock returns; and, (b) whether our three-factor model findings can be

explained by the turn of the year effect.

Our findings suggest that idiosyncratic volatility is highly correlated with firm size and

that it is useful in explaining expected stock returns. In Tables 7 and 8 we present the

premia generated by market, firm size and idiosyncratic volatility for Germany and

United Kingdom. We find that small firms generate higher returns because they have

high idiosyncratic volatility. Thus, we argue that idiosyncratic volatility is correlated

with firm size. Interestingly, for UK we find that big firms have higher idiosyncratic

volatility and thus those portfolios generate superior returns. Hence, we advance the

argument that investors who invest in stocks with these characteristics tend to take

greater risk and thus higher risk premia are compensation for these risks. As far as

the seasonality issue is concerned we do not find any evidence of our results being

explained by the turn of the year effect. Our findings are consistent with Malkiel and

Xu (1997 and 2000) who find that idiosyncratic volatility is useful in explaining cross-

sectional expected returns. They also observe that idiosyncratic volatility is related to

the size of the firm in that small firms have high idiosyncratic volatility thus providing

an alternative explanation to the FF (1992) conclusions. Thus, we demonstrate that

idiosyncratic volatility plays an important role in empirical asset pricing. In closing, we

argue that the CAPM beta alone is not sufficient to describe the variation in average

equity returns.

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2

Figure 1.0 Mean Monthly Returns Germany

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

S/L S/M S/H B/L B/M B/H

Portfolios

Mon

thly

Ret

urns

(%)

3

Figure 2.0 Mean Monthly Returns United Kingdom

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4


Portfolios

Mon

thly

Ret

urns

(%)

4

Figure 3.0 Market, Size and Idiosyncratic Volatility PremiaGermany

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


Portfolios

Mon

thly

Pre

mia

(%)

MARKETSIZEIDIO RISK

5

Figure 4.0 Market, Size and Idiosyncratic Volatility PremiaUnited Kingdom

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5


Portfolios

Mon

thly

Pre

mia

(%)

MARKETSIZEIDIO RISK

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